There many ways

To start with go to google and type in your keyword. You will see suggestions for other related keywords. This method is free.

Kenj

https://freekeywords.wordtracker.com is another one. Its free for the basic service but if you want to drill down into the keywords that came up with there is a cost.
You could use the keywords you come up with to put in Google and see what the number results show or what new keywords it might bring up that you wouldn't have had before.

the google wonder wheel is a very good and unused way to do keyword research.
you can also drill down using topics and keywords.

yup, Google adwords keyword tool, and doing a simple Google search of your niche field are my favorite free and easy methods. You should definitely have a look at Wordtracker, their free service is good but having to pay to see more kinda always bothered me.

It is simple. Google Keyword Tool will give you similar keywords and will also give you statistics for those keywords.